Equilibrium Wind Hedge Contract Structure through Nash Bargaining

Transcription

1 Proceedings of the 2011 Industrial Engineering Research Conference. Doolen and E. Van Aken, eds. Equilibrium Wind Hedge Contract Structure through Nash Bargaining Harikrishnan Sreekumaran and Andrew L. Liu School of Industrial Engineering, Purdue University West Lafayette, Indiana Abstract With the continued maturing of electricity markets, merchant wind project developers are increasingly looking to longterm hedge contracts to control their exposure to market risks as well as to secure project financing. Such contracts are usually negotiated with load serving entities looking to hedge against spot price volatility or financial firms who repackage the electricity through structured deals that create value. We model the negotiation process using the Nash bargaining framework. he equilibrium price and quantity come out as the optimal solutions of a stochastic program wherein the objective is to maximize the product of the utility differentials for the two parties, with and without the contract. We use conditional cash flow at risk, a risk based profitability measure, to capture the risk attitudes of the bargaining parties. After incorporating the risk measure into the utility functions, we solve the resulting stochastic program using the sample average approximation method. We present numerical results as well as sensitivity analysis for the equilibrium contract structure with respect to volatility and correlation between a wind plant s outputs and spot electricity prices. We also study how the risk attitudes of the two parties affect the equilibrium structure. Keywords Wind energy, hedging contract, Nash bargaining, stochastic programming, risk management. 1. Introduction Growing economic and social pressures in recent times have contributed to the prominence of renewable energy sources as an alternative to conventional electricity production. he last decade has seen a significant increase in wind energy production, especially in developed nations. he key advantages of wind power include that it is essentially free of fuel costs and air pollutant emissions. herefore generating wind power generally has low operating costs. However wind power projects are capital intensive, which makes risk management a key issue. Wind energy producers have to deal with the inherent uncertainties present in the generation capacities as a result of variation in wind-speed as well as the financial risk caused by electricity spot price volatility. o obtain sufficient capital financing for projects, producers are often required to enter into long-term contracts with credit-worthy power purchasers for the purpose of out-sourcing the risk. he contracts may take the form of long-term power purchase agreements in which the wind farm sells a certain fixed quantity to an electricity retailer at a fixed price over a certain period of time. Such contracts involve physical delivery of electricity which may be complicated by the transmission process. he wind farm may also resort to purely financial Contract-for-Difference (CFD) hedges in which the two counterparties agree to a fixed quantity, strike price and maturity period. Once the spot-market price of electricity is known the parties settle the difference between the strike price and the spot price amongst themselves. Either type of contract may include the sale of Renewable Energy Certificates (RECs) as well. Since there is no actual transfer of electricity involved in a CFD hedge, such contracts may be entered into not just with electricity retailers but also other parties such as investment firms. he viability of such long-term hedging contracts hinges on the difference in risk-aversion between the wind power producer and the power purchaser. In this context it becomes important for both parties to analyze the risk-return implications of the contract so as to arrive at optimal contract parameters that maximizes return while restricting the

2 risk involved to acceptable levels. Although the problem of risk management in the context of electrity derivates has received some attention, not much work has been done in the bargaining and negotiation framework. An iterative negotiation scheme for bilateral electricity contracts, which enables the counter parties to reach mutually beneficial and risk tolerable contract payoffs was proposed in [1]. Nash bargaining theory was used in [2] to derive contract structures for nonstorable commodities. However the focus there was on a mean-variance based risk analysis, which has severe limitations as we shall explain. A similar model for studying forward contract negotiations in the electricity market was used in [3], where cross market effects were analyzed. In this study, we analyze the risk management aspect of a long term CFD contract between the wind farm and a buyer. We model the negotiation process as a Nash Bargaining game [4]. We model the risk preferences of the two parties using the Conditional Cash Flow at Risk (CCFaR) risk measure [5, 6]. he equilibrium solution to the bargaining game is characterized by a stochastic optimization problem which we solve using the sample average approximation (SAA) method [7, 8] to deal with the underlying uncertainties. he remainder of the study is organized as follows. Section 2 describes the model we use for the negotiation process and gives a brief review of the utility functions we use. he formulation of the Nash bargaining optimization problem is given in Section 3 along with certain reformulations to make the problem more tractable. An algorithm based on the SAA method is given in Section 4. Some preliminary computational experiences are given in Section 5 and the study is concluded in Section 6 along with planned future extensions of this work. 2. Contract Model We consider a CFD contract between the wind farm (the seller) and a buyer. he two parties agree to a maturity period, and negotiate a fixed strike price and quantity pair. Once these contract parameters are agreed upon, the wind farm then sells its energy ouput to the spot market. Each month, if the average spot price is above the agreed upon strike price, then the wind farm compensates the buyer for the difference. Conversely, if the average spot price is below the strike price, then the buyer will pay the difference to the wind farm. he data and parameters associated with the model are given below. Input Data he maturity period of a hedge contract, in months. ω Random variable defined on a probability space (Ω,F,P) of the underlying uncertainties. K t (ω) Capacity of a wind power plant at time t with t ; [MW]. p t (ω) Wholesale electricity price at time t with t ; [$/MWh]. r Risk-free discount rate. δ t Discount factor at time t; δ t = 1/(1 + r) t. λ B, κ B Measures of the buyer s risk aversion. λ S, κ S Measures of the seller s risk aversion. Variables and Functions p c q c Contract price [$/MWh]. Contract quantity [MW/month]. πc B, πb NC he payoff function of the buyer with/without entering a contract; [$]. πc S, πs NC he payoff function of the seller with/without entering a contract; [$]. U B (π B ) he utility function of the buyer. U S (π S ) he utility function of the seller. Assuming that the wind farm sells all of its actual energy output on the spot market, its profit is then the earnings from the energy sales combined with any earnings realized from the hedging contract. he profit for the buyer is simply any

3 earnings it realizes from the difference in the spot price and the strike price. he profit terms for each party can then be given as π S C (pc,q c ) = = = π B C (pc,q c ) = δ t [p c q c + (K t (ω) q c )p t (ω)] δ t [p c p t (ω)]q c + δ t [p c p t (ω)]q c + π S NC ; δ t [p t (ω) p c ]q c. δ t K t (ω)p t (ω) Since the difference in risk aversion between the two parties involved is of paramount significance, it is important to model the preferences of the two parties appropriately. As a starting point it is natural to model the risk aversion using mean-variance preferences. In this case the utilities for the two parties is (1) U S (π S C ) = E(πS C ) λs var(π S C ) = E δ t [p c p t (ω)]q c + E(π S NC ) [ ] [ ]} λ {var S δ t q c p t (ω) + var(π SNC ) 2cov π S NC, t q δ c p t (ω) ; (2) U B (π B C ) = E(πB C ) λb var(π B C ) = E δ t [p t (ω) p c ]q c λ B var { δ t [p t (ω) p c ]q c } Although variance has been a popular risk measure ever since Markowitz s seminal work on mean-variance portfolio selection, an issue is that variance punishes both excessive profit and excessive loss, which is not desirable. One of the popular down-side risk measures that addresses this issue is Value at Risk (VaR), which denotes the threshold level over which the loss on the portfolio is unlikely to go at some given confidence level. However, being a quantile function, VaR cannot account for low probability high risk events which are especially troublesome with fat-tailed loss distributions. Artzner et al. [9] provided an axiomatic approach defining the desirable properties for a risk measure to be a good quantifier for the risk of loss. hey characterized coherent risk measures using the properties of sub-additivity, monotonicity, translation invariance, positive homogeneity and relevance. Rockafellar [10] described a slightly different characterization of coherent risk measures, in terms of the additional property of convexity, which becomes significant in the optimization context. It was also shown that coherent risk measures preserve convexity in the sense that if the loss function is convex in the decision space, then so is the risk measure. It has been shown that VaR is coherent only if it is based on normal loss distributions. For general loss distributions VaR lacks sub-additivity and convexity. VaR can also be ill behaved as a function of portfolio positions, and can exhibit multiple local minima. o address this issue and to account for fat-tailed loss distributions, we choose Conditional Cash-Flow at Risk (CCFaR) [6] as our risk measure. he idea of CCFaR arises from the risk measure Conditional Value at Risk (CVaR) proposed by Uryasev and Rockafellar [5, 11]. CVaR is defined as the conditional expectation of the losses in excess of the Value at Risk at a given confidence level. It is also sometimes known as mean excess loss, tail VaR and mean shortfall. Another reason for not using VaR in our work is that there is almost no market for trading the wind hedge contracts once they are signed. An (implicit) assumption of using VaR as a risk measure is that any risky position can be closed at any time, which clearly does not hold in our case. With all the justifications discussed above, we choose CVaR as our risk measure, and extend the measure from the value domain to the cash-flow domain, which is then the the.

4 Conditional Cash-Flow at Risk (CCFaR). he expression for CCFaR (and Cash-Flow at Risk, which is needed in defining CCFaR) with a given confidence level α is as follows. Consider the utility function given by CFaR α = inf{γ : Prob(Cash Flow γ) 1 α}; CCFaR α = E [Cash Flow Cash Flow CFaR α ]. U(π) = µ(π) + κ CCFaR(π), where µ(π) is the expected profit and CCFaR(π) is the conditional cash flow at risk. Note that we use of a different risk aversion factor κ than that in the mean-variance formulation (λ) to indicate that the two risk-aversion factors are not the same in general. Note that the estimation of risk-aversion factors is beyond the scope of this paper, and will be addressed in future research. he utility differential for the seller can then be given as U S = U S (π S C (pc,q c )) U S (π S NC (pc,q c )) = E Similarly the utility differential for the buyer can be given as δ t [p c p t (ω)]q c + κ S [ CCFaR(π S C (pc,q c )) CCFaR(π S NC (pc,q c )) ]. U B = U B (π B C (pc,q c )) U B (π B NC (pc,q c )) = E 3. Nash Bargaining Formulation δ t [p t (ω) p c ]q c + κ B [ CCFaR(π B C(p c,q c )) CCFaR(π B NC(p c,q c )) ]. We model the negotiation process between the wind farm and the buyer as a Nash bargaining game. he two parties seek to agree on a mutually beneficial utility pair (U S,U B ). Instead of modeling a sequential bargaining process, Nash [4] listed four axioms to be satisfied by an equilibrium from a bargaining process, and showed that the bargaining problem has a unique solution satisfying the axioms if and only if the payoffs to the players are as follows. π := (π 1,...,π F) arg max F d π Γ f =1 where ν = (ν 1,...,ν F ) with ν f being player f s payoff should the bargaining fail, and Γ = (π f ν f ), (3) F f =1 Γ f with Γ f being the set of player f s feasible payoffs. he Nash Bargaining solution (NBS) was shown [4, 12] to be unique under the assumption that the feasible utility set for the two parties is a non-empty, compact, convex subset of R 2. Nash bargaining theory is extended to non-convex problems and various relaxations to the axioms were explored in [13, 14, 15]. Given the utility functions from the previous section, we can now formulate the Nash Bargaining problem as maximizing the product of the utility differentials for the two parties over the feasible regions of p c and q c so long as the utility differentials for both parties are non-negative. Let Ψ S (p c,q c ) = E Ψ B (p c,q c ) = E δ t [p c p t (ω)]q c κ S CCFaR(π S NC (pc,q c )), and δ t [p t (ω) p c ]q c κ B CCFaR(π B NC(p c,q c )).

5 Using this notation we can now formulate our NBS problem as [ maximize i {S,B} Ψ i (p c,q c ) + κ i CCFaR(π i p c, q c 0 C (pc,q c )) ] subject to Ψ i (p c,q c ) + κ i CCFaR(πC i (pc,q c )) 0 for i {S,B}. (4) he CCFaR terms in the problem contain conditional expectation terms which are difficult to work with. he key to deal with these is their relationship with the following auxiliary functions. Γ S (p c,q c,τ S ) = τ S 1 1 α E[τS π S C (pc,q c )] + ; Γ B (p c,q c,τ B ) = τ B 1 1 α E[τS π B C(p c,q c )] +, where [ ] + is defined as maximum between the argument in [ ] and 0. It can be shown that instead of dealing with the conditional expectation terms in CCFaR, we may replace the CCFaR terms with the auxiliary function terms which are then maximized over the additional τ i variables as well. o show this results, we first use a version of heorem 10 from [11] which gives a characterization of CCFaR in terms of the auxiliary function. Lemma 1 Under the assumption that the probability distribution for the underlying uncertainty factors is continous, Γ i (p c,q c,τ i ) is a convex, continuously differentiable function in τ i. he α-ccfar of the profit associated with a feasible (p c,q c ) pair can be determined from the formula CCFaR α (π S C (pc,q c )) = max τ i Γ i (p c,q c,τ i ). In this formula, the set consisting of the values of τ i for which the minimum is attained, namely A α (p c,q c ) = argmin τ i Γ i (p c,q c,τ i ), is a non-empty closed bounded interval (perhaps reduced to a single point), and the α-cfar of the profit is given by CFaR α (p c,q c ) = left end point of A α (p c,q c ). Proof his is exactly heorem 10 in Rockafellar and Uryasev [11] with CVaR replaced by CCFaR. Proposition 2 Solving (4) is equivalent to solving the following problem. [ maximize i {S,B} Ψ i (p c,q c ) + κ i Γ i (p c,q c,τ i ) ] p c, q c 0, τ S, τ B subject to Ψ i (p c,q c ) + κ i Γ i (p c,q c,τ i ) 0 for i {S,B}. (5) Proof Using Lemma 1, we can replace the CCFaR terms in (4) with the auxiliary function as follows ] maximize i {S,B} [Ψ i (p c,q c ) + κ S maxγ i (p c,q c,τ i ) p c,q c 0 τ i subject to Ψ S (p c,q c ) + κ i maxγ i (p c,q c,τ i ) 0 for i {S,B}. τ i Now since the Ψ i (p c,q c ) is independent of τ i, we can take the inner max term out of the objective function. We can also remove the max terms in the constraints. his is easily seen since even if some τ i / argmax{γ i (p c,q c,τ i )} that satisfies the constraints, it certainly will not be optimal to our problem. his leaves us with problem (5). At this point we can apply discretization techniques, as used in [5], to the expectation terms in Γ i (p c,q c,τ i ). Given a sample of N spot price and capacity realization pairs (p n,q n ), the expectation term in the auxiliary function may be approximated by E[τ i π i C (pc,q c )] + = 1 N N n=1 [τ i π i C (p n,q n )] +.

6 Using this discretization and using dummy variables for the positive part function, we obtain the following final approximation to (5) maximize v S v B p c,q c,v B,v S,τ B,τ S ] subject to v S Ψ S (p c,q c ) + κ [τ S S N 1 N(1 α) z S n n=1 ] v B Ψ B (p c,q c ) + κ [τ B B N 1 N(1 α) z B n n=1 z S n τ S πc S (p (6) n,q n ) for i = 1,...,N z B n τ B πc B(p n,q n ) for i = 1,...,N p c,q c,v S,v B 0 z B n,z S n 0 for i = 1,...,N. 4. Sample Average Approximation Solving problem (5) involves evaluating the expectation terms present in Γ i (p c,q c,τ i ). For general loss distributions, explicit analytical expressions for these expectations are rarely available. However, the discretization process used to arrive at problem (6) can be thought of as an application of Sample Average Approximation (SAA) [7] to overcome this difficulty. SAA is a well known method for dealing with expectation terms in stochastic programming. he idea is to generate a random sample and solve a deterministic problem using the sample means to estimate the expectations. Under mild assumptions it has been shown that an optimal solution and the optimal value of the SAA problem converges exponentially to their true counterparts as the sample size increases. Analyses of the convergence rates and statistical validation schemes for SAA applied to expected value constrained stochastic programming problems can be found in [16, 8]. Convergence results for the SAA method were extended to the domain of non-convex stochastic programming problems in [17]. We propose here an SAA based algorithm to solve problem (6). Observe however that the terms Ψ i (p c,q c ) are separable in terms of the decision and random variables, while we deal with the CCFaR terms not by separation but by discretization and linearization. his is an important distinction since the separation process in the former simply involves evaluating certain statistics based on the sampling while the latter involves adding a variable and a constraint for each sample involved. So the sampling is used in two completely different ways for the two terms - one merely for computation of certain statistics, the other for reformulating the optimization problem. We therefore consider using a 2-stage sampling process, as the number of samples desired for the evaluation in the first stage might be different from that desired for the linearizations in the second stage. he proposed algorithm is given below. Algorithm 3 2-Stage SAA for Problem (5) 1. Choose appropriate sample size(s). 2. Stage 1 - Based on the samples, evaluate the appropriate terms and obtain the polynomial coefficients for the decision variables in Ψ i (p c,q c ) terms. At this stage we have Problem (5) 3. Stage 2 - Perform the discretization and linearization based on the sampling to obtain Problem (6) 4. Solve Problem (6) with a nonlinear programming (or global optimization) solver. 5. Perform validation using extensions of the SAA bounding techniques. 5. Preliminary Numerical Results Preliminary numerical results are obtained for a test case of a wind farm with 30 MW generation capacity. he spot electricity price data is generated using a Brownian-motion model and the NBS equilibrium price and quantity pairs are obtained using enumeration on the feasible space to solve problem (4). Comparative statics for varying risk aversion factors for the buyer and varying price volatilities are given in Figures 1 and 2 respectively.

7 It is seen that as the buyer becomes less and less risk averse which is reflected by a lowering of the risk factor κ B, they are willing to pay larger prices for the electricity for larger quantities. his results in larger profits and less risk for the wind farm. On the other hand, increase in price volatility drives the contract price down. his makes intuitive sense because as the spot price becomes more volatile the buyer is taking on more risk. herefore they demand a higher premium for the risk in terms of a lower contract price. Figure 1: Comparative Statics for Risk Aversion Factors 6. Conclusions Figure 2: Comparative Statics for Spot Price Volatility In this work we model the negotiation process for Contract-for-Difference (CFD) hedges between wind farms and buyers as a Nash bargaining game. he equilbrium solutions to the game can be obtained through solving a stochastic programming problem. Preliminary numerical results are presented to validate the model. A Sample Average Approximation (SAA) based algorithm is proposed to solve the optimization problem to obtain the equilbrium contract structure. Further work needs to be done on deriving the conditions under which the Nash Bargaining Solution is unique. Solution methodologies will be developed to solve the NBS optimization problem keeping in mind its non-convex nature. Since the non-convexity is caused by bilinear terms, decomposition methods such as Benders decomposition will be studied. he convergence implications of such algorithms within the SAA framework remains to be analyzed. he model used here may be extended to more general hedges in the wind industry. he immediate generalization involves adding RECs into the mix. he general framework presented here may be used for analyzing the risk implications in many other kinds of bilateral contracts.

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